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REVISED AND ENLARGED. 



NOTES 



ON THE 



Jirsi | ook of | mm* |^omdrg, 



AND 



CONCERNING THE CIRCLE, 



SHOWING 



The DECIDED IMPROVEMENT 



EFFECTED IN 



The Science of Geometry. 



BY 



LAWRENCE S. BENSON, 

author of " benson's geometry." 



NEW YORK : 

James S. Burnton, Publisher, 149 Grand Street, 

1873. 



Entered according to Act of Congress, in the year 1873, by Lawrence S. Benson, in the 
office of the Librarian of Congress, at Washington. 



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NOTES 



ON THE 



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Jksl look of iWsim'a |)«om^rg, 



AND 



CONCERNING THE CIRCLE ; 



SHOWING 



The DECIDED IMPROVEMENT 



EFFECTED IN 



The Science of Geometry. 



BY 



\r- 



LAWRENCE S. BENSON, 

author of "benson's geometry." 



NEW YORK, A 

James S. Burnton, 149 Grand Street 
1873. 




PREFACE. 

The object of this treatise is to institute rigid comparisons be- 
tween the definitions, axioms, postulates and demonstrations of the 
First Book of Benson's Geometry, and the similar definitions, 
axioms, postulates and demonstrations given by other geometers. 
It is to point out the various changes and improvements effected 
by Benson's Geometry, and it is to assist teachers in imparting 
to beginners the rudiments of the science. The author hopes to 
make the study of geometry so easy, and free from difficulty, that 
learners will readily comprehend the scope of geometrical science, 
and clearly understand the purport of geometrical reasoning. 

The importance of scientific education cannot be over-rated. 
For science is the key by which we unlock the mysteries of Nature, 
and behold the vast Universe in all its naked realities. It is more, it 
is the great light by which we can penetrate the darkness of the 
far-distant Past, understand the Present, and fathom the eternity of 
the Future. Science teaches us the wisdom and power of God. 
Theie is as much systematic skill displayed in the creation of an 
animalcule as there is in the construction of a planet. The 
vitality of the one is as grand as the existence of the other. The 
power which condenses the fluid is the same as that which gives 
form and shape to the universe. Science is the embodiment of 
wisdom and power. God is the great fountain-head of science. 

Knowledge is progressive. When we compare the extent of the 
modern sciences with the scanty acquisitions of the ancients, the 
great improvements and advancement of modern researches as- 
tound the mind more than all the astrological signs and magical 
sways of the oracles and other depositories of ancient lore. The 
mind of man was in that period of mental poverty hampered with 
materiality, and was filled with superstition and dread. Its move- 
ments were attended with cautiousness and distrust. It was when 
facts had multiplied, and the genius of man awoke and discovered 
the orders of relations, the traces of resemblances, the points of 
contrasts, and the lines of connections between them, that the 
mind claimed the rights of intellectuality. As ages follow ages, 
the field of the intellects become more and more extended, and 
the province of the mind grows richer and grander. 



4 PREFACE. 

Mathematics is the most abstract of the sciences. There the 
intellects have full play. Though abstract it is not abstruse. Its 
principles are obvious, its demonstrations are rational, and its con- 
clusions are useful. Mathematics shows the superiority of educa- 
ted and intellectual faculties. Mathematics gives a grandeur and 
sublimity to the mental acquisitions. The remotest star is within 
its reach, and the immensity of the universe under its cognizance. 

It has frequently been inquired : What is the use of Geom- 
etry ? It is surprising that a science so useful and so long estab- 
lished as Geometry has been, should be so little known by the 
general mass of mankind. The celebrated philosopher, Plato, is 
said to have put an inscription over the door of his school, stating 
that no one unacquainted with Geometry should enter it. The 
joy of Pythagorus over his discovery of a certain geometrical 
proposition, is said to have been so great that he sacrificed a hun- 
dred head of oxen in honor of the Muses. 

Ptolemy Philadelphus, King of Egypt, wished to know if 
there was an easy or short way of learning Geometry. Euclid 
replied, "No, Sire, there is no royal road to Geometry." The 
famous city of Alexandria, in Egypt, founded by Alexander the 
Great, was made the royal residence of Ptolemy Lagus, King 
of Egypt, who established the celebrated Museum there, which 
continued to be the chief seat of Mathematics for nearly one thou- 
sand years, till it was destroyed by the Saracens in the seventh 
century of the Christian era. Pages could be rilled with accounts 
of the interest which Geometry has awakened in the minds of the 
most distinguished and eminent men of the world ; but a few facts 
showing the great practical importance of Geometry will suffice. 
The lawyer has a difficult case before court : a human life depends 
upon his pleadings. His client is innocent, but circumstantial 
evidence is strong against him. The lawyer sifts the evidence, 
argues upon it, and saves his client. Was it witchcraft ? No ! 
The lawyer was well drilled in geometrical reasoning — his argu- 
ments were clear, logical and sound, and his conclusion was valid, 
evident and indisputable. The present advancement and progress 
of human achievments are wonderful. Commerce empties her 
cornucopia into the lap of mankind. Enterprise rears magnificent 
structures, builds stupendous bridges, and throws networks of 
railroads and telegraphs over the habitable globe. Science maps 
out the firmament, traces the paths of the erratic comets, foretells 
eclipses, the rising and setting of the sun, periodic phases of the 
moon, the rise and fall of the tide, and prognosticates the atmos- 
pheric changes with almost absolute certainty; These things are 
all dependent upon geometrical principles and geometrical truths. 



NOTES ON THE DEFINITIONS. 



THE POINT. 

The definition of a point given by Euclid, viz : " That whtck 
has no parts or zuhich has no magnitude" has been objected to by 
geometers because, as Playfair says, it is negative only, and is 
not convertible, for everything that has no parts or is without mag- 
nitude is not a point ; and Playfair changes the definition to the 
following, viz : "A point is that which has position, but no magni- 
tude" and he claims that this definition, in its affirmative part, in- 
cludes all that is essential to a point, and that its negative part 
includes everything that is not essential to it. See notes to Play- 
fair's Euclid. Subsequent geometers have unreservedly adopted 
Playfair' s definition. 

In Benson's Geometry I have thought proper to deviate from 
Playfair's definition in the following manner, viz : " Geometers 
define & point, position without magnitude ; but to give a point 
position, would entitle it to the three dimensions of magnitude ; 
whereas a point in Geometry expresses no dimension." By which 
it will be seen that I adhere closely to Euclid's definition. The 
notion that we acquire of a point is, that it has no properties, which 
is essentially negative ; therefore, the definition must also be neg- 
ative; because, when we acquire a notion which is essentially neg- 
ative, the term by which that notion is expressed must necessarily 
be negative ; that is, our definition of a thing must correspond with 
our notion of it ; for it is impossible to give an affirmative definition 
of a thing when our notion of it is negative. This holds true in 
regard to Playfair's definition, his notion of a point is negative, 
and he is obliged to define a point as having no magnitude. But 
he gives a point position. Now position involves situation, and 
situation involves existence, and whatever exists must have proper- 
ties ; whereas a point has no properties, consequently it can have 
no existence, and having no existence it can have no position. 
When we give position to a thing, we treat it as an existing thing, 
having magnitude — hence, however that Euclid's definition may 
be " not convertible," Playfair's definition is inconvertible, for 
it is an anomaly — it confounds an affirmation with a negation ; 
that is, it asserts a thing and denies it at the same time. Euclid 
having acquired a negative notion of a point, he had to give it a 
negative definition, and when a geometer acquires an affirmative 
notion of a point, then he will be able to give it an affirmative 



6 NOTES ON THE DEFINITIONS. 

definition. Therefore, when our notion of a point is negative, the 
negative definition is perfectly valid, consistent and rational. 

Playfair, in his second note, states that the termination of a 
line has no magnitude, but it has necessarily position, hence it is 
a point. Is it possible to conceive of position without magnitude ? 
I think not. Although we regard the termination of a line, or the 
intersection of two lines, as a point. It is not because the term- 
ination or the intersection has anything essential to a point ; for 
the termination or intersection is no more a point than is a mark 
on the paper a line, or a dot a point. The termination or the in- 
tersection of marks on paper is a material representation ; but a 
geometrical point, and a geometrical line, are abstractions, and 
although it is agreed to call the termination of a mark or the in- 
tersection of two marks a point ; still, either is simply a material 
representation of the geometrical point, and it is not reasonable to 
draw a geometrical inference from material things. When we say 
that the termination, or the intersection of marks on paper, is 
a point, we do not mean that either is actually so, because, draw 
marks ever so fine, they will have width, consequently, they are 
not geometrical lines ; and because we can give position to a mark 
or a dot, it does not follow as a necessary consequence that we 
can give position to a geometrical line or a geometrical point 
For such reasoning would reverse the whole order of geometrical 
inquiries, which are to derive practical conclusions from abstrac- 
tions, and not vice versa. 

That the geometrical point expresses no dimension is shown in 
the fact that all geometrical magnitudes have dimension. The line 
has length, the surface has length and width, and the solid or 
volume has length, width and thickness. The point being no 
magnitude, consequently it has no dimension. The dot, extremity 
of a mark, and the intersection of marks upon paper, have position, 
Out tiie geometrical point is an abstraction of the mind, and 
certainly it can have no position. 

THE STRAIGHT LINE. 

Geometrical principles are essentially metaphysical, because 
geometrical properties are so entirely mental abstractions that the 
fundamentals of geometrical science can be treated in a meta- 
physical point of view only. Euclid, in defining a straight line, 
considered it as a line that lies equally between its extreme points, 
that is in reference to the space on each side of the line. But the 
objection to this is that it is "too metaphysical ! " However this 
may be, it is certainly not strictly a definition. The word 
" equally," or as some translators render it, " evenly," embodies 
something to be proved. Hence, Euclid's definition is faulty, 



NOTES ON THE DEFINITIONS. 7 

which he seemed to have been aware of himself; for he never at- 
tempted to deduce from it any property of the straight line ; but 
he substituted in its stead the axiom that two straight lines cannot 
enclose a space, and all his demonstrations concerning the straight 
line are founded upon the axiom. Playfair, to overcome the ob- 
jections to Euclid's definition, has substituted the following, viz : 
If there he two lines which cannot coincide in two points without 
coinciding altogether, each of them is called a straight line." His 
definition is liable to the same objection as Euclid's. A de- 
finition is required to be so clear, simple and complete that there ' 
can be nothing desired to make it more so. Now Playfair says 
that " if there be two lines which cannot coincide in two points 
without coinciding altogether, each of them is called a straight 
line." Is it possible to have two distinct lines " coinciding al- 
together ? " Can the human mind conceive such a possibility ? 
Yet Playfair infers as much when he says "if there be two lines 
which cannot coincide in two points without coinciding altogether, 
each of them is called a straight line." Again, when we have 
two lines coinciding in two points, it is a matter to prove that those 
lines are straight because they cannot coincide in two points with- 
out coinciding altogether. The illustration of Playfair' s de- 
finition would be to draw a line, then draw another line so as to 
coincide with the first line in two points, then if the two lines 
" coincide altogether," each of them is a straight line, according 
to Playfair. Now, let us draw a curve line ; and we can draw 
another curve line exactly upon the first curve line, which will 
coincide with the first curve in two points, the extremities, and 
which will " coincide altogether " with the first curve line. Hence 
we have fulfilled the conditions of Playfair's definition, but we 
have no straight lines. 

Playfair says again : U A straight line is one in which, if the 
position of two points be determined, the position of the whole line is 
determined." Now, why cannot this be said of the circumference 
of a circle ? Of the curve of a parabola ? Of the curve of an 
ellipse ? Or of the curve of an hyperbola ? 

I have dwelt upon Playfair's definitions because they have 
been adopted unreservedly by all subsequent geometers. 

In Benson's Geometry, I have deviated from Euclid's de- 
finition, and also from Playfair's. I have given the following : 
" A line expresses length only, and is capable of two conditions — 
it can be straight or curved ; when its length is always in one 
direction, it is straight ; but when there is a continual variation 
in the direction of its length, it is curved ; or in brevity called 
a curve." 

I have not put any encumbrances in my definition. I have said 
that a line expresses length only. I then state it is capable of 



S NOTES ON THE DEFINITIONS. 

two conditions, namely, a line can be straight or curved. I ac- 
cordingly define what condition is necessary to a straight line, and 
also, what condition is necessary to a curve. The idea that a 
straight line must have its length in one direction, is simple, clear 
and complete ; and the idea that a curve line continually varies 
the direction of its length is also simple, clear and complete. A 
distinction is at once shown between the straight line and the 
curve ; and there is no danger that they will ever afterwards be 
confounded together. 

And what is of great importance, this definition has practi- 
cability; it can be used in the demonstration of various proposi- 
tions, which Euclid's definition cannot be. 

SURFACES. 

Euclid's definition of a plane surface was similar to his defini- 
tion of a straight line ; he defined a plane surface as one which 
lies equally or evenly between its extreme lines. It is open to the 
same objections that have been made to his definition of the 
straight line. 

Playfair gives this definition: "That those superficies are 
called plane, which are such, that if three points of the one 
coincide with three points of the other, the whole of the one must 
coincide with the whole of the other." The remarks I have made 
in reference to his definition of a straight line apply with equal 
force to this definition. We can conceive a spherical triangle 
formed on the surface of a sphere, as having the vertices of the 
angles coinciding with the vertices of the angles of another and 
equal spherical triangle formed on the same sphere, or an equal 
sphere — the two spherical triangles will coincide in three points and 
throughout, but the surfaces of the spherical triangles are not planes. 

Simson's definition is, " that a plane superficies is that in which 
any two points being taken, the straight line between them lies 
wholly in that superficies." This definition is very good so far as 
it goes ; but the object of a definition is to make a distinction, and 
we are not informed what a curved superficies is, except that it is 
not a plane superficies, nor composed of plane superficies. I have 
thought proper to deviate from Simson's definition in order to show 
readily the distinction between a plane and a curved surface. 

In Benson's Geometry, I give the following definition : " A 
plane surface, or something called a plane, is one in which any 
line can be drawn wholly in the surface ; an4 a curved surface is 
one in which a curve only can be drawn wholly in the surface in 
the direction of the curvature." 



NOTES ON THE DEFINITIONS. 



THE ANGLES. 



There is much ambiguity in the definitions by Geometers for 
angles. Euclid called an angle the inclination of two straight 
lines to one another ; another Geometer called it the divergence 
of two straight lines ; another Geometer called it the difference of 
direction of two straight lines. Prof. Davies, in his earlier edi- 
tion of Davies' Legendre, defined an angle as " a portion of a 
plane included between two straight lines meeting at a commor 
point." Now to call an angle " a portion of a plane " gives length 
and breadth to an angle, which is absurd, for it asserts that two 
straight lines can enclose a space. In the later edition of Davies* 
Legendre, the same notion is retained with a change in the phrase- 
ology — as follows : — " A plane angle is the amount of divergence 
of two lines lying- in the same plane." This is both indefinite and 
absurd, for the "two lines " may be curves in which case they will 
not form a plane angle and "two lines lying in the same plane" 
are not necessarily compelled to meet each other, hence two lines 
lying in the same plane will not necessarily form a plane angle ; and 
" the amount of divergence " of two lines conveys the notion of 
length and breadth to an angle, an absurdity that cannot be too 
emphatically denounced. 

In Benson's Geometry, I give this definition : " An angle 
is formed by two straight lines meeting each other, the point 
of intersection of the lines is called the vertex." This is simple 
and clear. 

There is also much ambiguity about right angles. Euclid and 
subsequent Geometers, in their definition, always suppose a certain 
condition understood which their definition will not warrant. 
They say that when two straight lines meet, the adjacent angles 
formed by them are right angles, when they are equal. Now two 
straight lines, on meeting, are not necessarily compelled to make 
adjacent angles- The lines can meet and make only one angle. 
Therefore, in Benson's Geometry, I give the following : " When 
one straight line meets another straight line, so as to make two ad- 
jacent angles, these angles are right angles when they are equal." 
There is also much ambiguity about the obtuse and acute angles. 
Geometers simply define them as respectively greater and less 
than the right angle ; but they fail to give any standard bywhich 
their magnitudes may be determined. After the above definition 
of right angles, I add : " and when one angle is greater than the 
other angle, the greater angle is an obtuse angle, and the less angle 
is an acute angle." Hence, by my definition, the magnitude of 
the adjacent angles is the standard by which we determine the 
right, obtuse and acute angles. 



IO NOTES ON THE DEFINITIONS. 



THE CIRCLE. 



The object of a definition is not only to show a distinction, but 
the terms by which it is expressed should also be perspicuous, 
complete and concise. 

Playfair, Simson and other Geometers use two or more . de- 
finitions to convey their idea of a circle. In Benson's Geometry, 
I use but one definition only. I give the following : " A plane 
surface contained by one line is a circle when every part of the 
line is equally distant from a point in the surface ; the point is the 
centre of the circle and the line is the circumference" 

THE AXIOMS. 

Euclid gives twelve axioms for Geometrical reasoning. Play- 
fair gives eleven axioms. Legendre gives five axioms. In 
Benson's Geometry, though giving all the axioms of Euclid, I 
have shown that they can be embraced by two axioms only, viz : 
Things which are equal to the same are equal, and things which are 
not equal to the same are unequal. Because when we use equals 
to add, subtract, multiply and divide, the equality in each case is 
not destroyed ; hence we have always an equality. And when 
we use unequals to add, subtract, multiply or divide, we have in 
each case inequality. And magnitudes which coincide in every 
respect are equal. A whole and a part are not equal to the same ; 
hence, are unequal; while a whole and all its parts are equal to 
the same, hence, are equal. Since, from the definition of right 
angles we have an equality between them, and since all right 
angles agree with the definition, all right angles are equal to the 
same thing ; hence, are equal to one another. Thus it is seen 
that the axioms necessary for Elementary Geometry can be 
embraced by two axioms only. 

the postulates. 

Postulates are requisite for geometrical demonstration, even as 
much so as axioms or definitions. Some geometers have not 
given any, but Prof. Charles Davies, in his " Davies' Legendre" 
has erred on the other extreme ; he has given too many, and has 
actually given as postulates propositions which are demonstrable, 
and which he even demonstrates in his third book. In Benson's 
Geometry, I have given the three postulates of Euclid, which I 
consider are necessary for geometrical reasoning. 



NOTES ON THE PROPOSITIONS. 

PROPOSITION I. 

The first proposition of Benson's Geometry is different from 
the first proposition in any other Geometrical work. This pro- 
position. is an easy problem, viz.: to describe an isosceles triangle 
on a finite straight line given in position. The learner who has 
acquainted himself with the preceding definitions, will have no dif- 
ficulty in solving it. And the two corollaries which accompany 
this proposition, present in a clear manner the relations existing 
between the equal angles and equal sides of isosceles triangles, 
and make easy what in other geometries are often difficult to 
learners. 

PROPOSITION II. 

This is the first proposition of Euclid, and the corollary which 
accompanies it in Benson's Geometry, presents in a clear 
manner, also, the relations existing between the equal angles and 
equal sides of equilateral triangles, which are not so forcibly 
presented in any other geometry. 

PROPOSITION III. 

This proposition is to prove the equality between two triangles 
which have two sides of the one equal to two sides of the other, 
and an angle in the one equal to an angle similarly situated in the 
other. The enunciation of this proposition in Benson's Geom- 
etry is different from the enunciation given in other text 
books of geometry, and it is made to embrace four cases, viz : 
first, when the equal angles are contained by the respectively 
equal sides ; second, when the equal angles are opposite to one 
pair of the respectively equal sides ; third, when the equal angles 
are opposite to the other pair of the respectively equal sides ; and 
fourth, the limitation that when the least sides respectively of the 
triangles are equal, and the angles opposite to them are equal, the 
angles opposite the greater respectively equal sides, must be ot 
the same kind, either both acute or both not acute. 

The reason of the change in the enunciation of this proposition 
is obvious. The geometer should give general truths and deduce 
particular cases from them; but in this proposition, geometers, 
from Euclid down to the present time, have given the first case 
only ; but it is well for learners to know that geometrical de- 
monstrations are based upon general principles which are true 



12 NOTES ON THE PROPOSITIONS. 

under all conditions and in every circumstance. The fourth case 
of this proposition may be claimed to be demonstrated in the 
sixth book of Euclid, but instead of the triangles having equal 
sides respectively, they have proportional sides, and it is shown, 
when the angles opposite to one pair of the homologous sides 
are equal, and those opposite the other pair are either acute 
or not acute, the angles contained by the proportional sides are 
equal. 

Geometers always allude to the fourth case, and they call at- 
tention to the limitation that the angles opposite to one pair of 
the equal sides must be either both acute, both right angles or 
both obtuse angles. But no where is it expressly stated that when 
the least sides of two triangles are respectively equal and when 
the angles opposite them are equal, that the angles opposite to the 
other pair of respectively equal sides must be of the same kind, 
either both acute or both not acute, in order to prove the equality 
of the triangles. 

I have given my demonstration of the first case somewhat dif- 
ferent from the demonstration of it by Euclid and other geom- 
eters, in order to make it more easy to the learner. 

proposition ix. 

The enunciation of this proposition is different from that given 
by Euclid and other geometers. It reads as follows : " When 
one straight line meets another straight line and forms two un- 
equal angles on the same side of that line, the two angles will be 
equivalent to two right angles." In this enunciation I expressly 
claim two unequal angles to be formed, for a straight line can 
meet another straight line and form one angle only, or it can form 
two right angles, then, in the latter case, there will be no neces- 
sity to prove their sum equal to right angles. And I have sub- 
stituted the word equivalent for the word equal, for two things are 
equivalent when they have equal magnitudes but do not coincide 
in all respects, which is the case of the two unequal angles and 
two right angles ; and things are equal when they have equal 
magnitudes and coincide in all respects, which is the case of all 
right angles. 

proposition x. 

This proposition is to prove that when, at a point in a straight 
line, two other straight lines on the opposite sides of it, make the 
adjacent angles together equivalent to two right angles, those two 
straight lines are in one and the same straight line, The de- 
monstration of this proposition by Euclid and all subsequent 
geometers, is attempted by the Reductio ad Absurdum. This 



NOTES ON THE PROPOSITIONS. 13 

species of reasoning is very often employed in Geometry, although 
it has been objected to by some able geometers ; but they have 
never enforced their objections with sufficient clearness and ef- 
fect as to produce its abandonment. The efforts made by mr 
have received general approval. That this species of reasoning 
is illogical can easily be shown ; its sophistry is shown in the 
following general manner : We have two magnitudes, A and B, 
they are either equal to each other or unequal to each other. By 
the Reductw ad Absurd 'um, it is first supposed that B is greater 
than A ; and B is assumed equal to a third magnitude, C, which 
is knozon to be greater than B ; hence a conclusion of absurdity is 
derived, from which it is inferred that because A cannot be equal 
to C, A cannot be equal to any magnitude greater than B. But, 
as C is not shown to exceed B by less than any assignable 
magnitude, there is no proof given why A cannot be equal to a 
magnitude less than C and greater than B. Nor is there any 
proof given to show that C exceeds B by some assignable 
magnitude equal to the excess of C over A. Similar remarks can 
be made in regard to the supposition that B is less than A. 

Now, in this proposition, B is a point in A B, where two straight 
lines, B C and B D on the opposite sides of A B, make the ad 
jacent angles A B C, A B D together equivalent to two right 
angles, then it is to prove that B D is the same straight line with 
B C. In the Reductio ad Absurdum, B D is supposed not to be 
the same straight line with B C, but that B E is. It is known 
that B E cannot be so — still an argument is based upon the sup- 
position that it is — and absurdity results, whence it is inferred that 
no other line except B D can be the same straight line with B C. 
But no proof is given to show why some line other than B E or 
B D is not the same straight line with B C. It is only because 
B E is not that line, that it is inferred that B D is the line — an 
inference, however, is no proof — hence, there is no proof by the 
Reductw ad Absurdum, that B D is the same straight line with B 
C. The demonstration given by me in Benson's Geometry, is 
as follows : C is a point in C D, where A C and B C on the op- 
posite sides of C D, make two adjacent angles, AC D and 
BCD, together equivalent to two right angles, then A C and B C 
are in one and the same straight line. At the point C, I erect a 
perpendicular C E, to A C. Then A C E is a right angle. Ac- 
cording to the hypothesis, A C D + B C D are equivalent to two 
right angles — hence *A C E -f- E C B are also equivalent to two 
right angles. Now, ACE being a right angle, E C B must also 
be a right angle and equal toACE; hence, C E is also per- 
pendicular to C B. Therefore, C E makes two right angles with 
A B, and is perpendicular to A B. But C E makes the same 
right angles with A C and B C ; hence, A C and B C are the 



14 NOTES ON THE PROPOSITIONS. 

same straight line with A B. Then, since C D makes with A C 
and B C two angles, A C D and D C B, equivalent • to two right 
angles ; and with A B the same two angles, A C D and D C B, 
A C and B C are again shown to be the same straight line with 
A B. This is a clear and simple demonstration, requiring no false 
premises, no circuitous nor sophistical argument, and it gives no 
absurd conclusion. 

PROPOSITION XIV.' 

This is the converse of the third proposition, and it has three 
cases ; first, when the respectively equal sides are between the 
respectively equal angles ; second and third, when the respectively 
equal sides are opposite to the respectively equal angles similarly 
situated. The enunciation of this proposition, like that of the 
third proposition, is expressed in general terms, so as to include 
the three cases ; in other Geometries the first case only is given. 
This proposition renders easy the direct demonstration of the 
proposition that: "If two angles of a triangle be equal to one 
another, the sides opposite to them are also equal to one another" 
for which all Geometers use the Reductio ad Absurdum. 

In Benson's Geometry, I have given its demonstration as a 
corollary to the fourteenth proposition, as follows : " Let the tri- 
angle, ABC, have the angle CAB equal to the angle C B A, 
then will A C be equal to C B. From the vertex C draw C D 
perpendicular to A B, then the angles C D A and C D B are 
both right angles, the angles CAD and C B D are equal by 
hypothesis, and the side C D common. Therefore, by the four- 
teenth proposition, the sides A C and C B are equal. This corol- 
lary is always attempted by the Reductio ad Absurdum, but how 
simple and clear is the direct demonstration ; and why should 
pupils be taught to reason from fallacies, use sophistical processes 
of reasoning, and derive absurd conclusions ? The direct reason- 
ing is more in accordance with the strictness of geometrical in- 
quiries ; is more agreeable to the nature of geometrical truths ; 
and approaches more to the spirit of geometrical science ; whereas 
the Reductio ad Absurdum is illogical and inconclusive. 

proposition xv. 

This is the proposition of Parallel Straight Lines, which have 
given great perplexity to geometers, and their demonstration has 
been attempted by the Reductio ad Absurdum. Since the Re- 
ductio ad Absurdum is not valid reasoning, the demonstrations 
given for this proposition being based upon the Redutio ad 
Absurdum, are not logical nor tenable. I have therefore given in 



NOTES ON THE PROPOSITIONS. 1$ 

Benson's Geometry, a demonstration for it, which is entirely free 
from the Reductio ad Absurdum. 

Bisect a straight line, A B, at C : erect a perpendicular, C D, 
to A B ; join A D and BD; if we apply the triangle A D C to 
the triangle D C B, so that the point A will be on D, the line 
A D on B D, and the triangles on opposite sides of D B, then 
D E and C B will be parallel straight lines, and will be equally 
distant from one another. Because D C is perpendicular to A B 
and bisects A B at C, the triangles A D C, B C D, are equal. 
On applying the triangles to each other, as above, we can 
represent the triangle A D C by D E B, then D E B is equal to 
BCD; hence they will have their sides and angles respectively 
equal. And since D E and C B are straight lines, they have no 
variation in the direction of their lengths between D and E, nor 
between C and B ; and because C D is equal to B E, D E and 
C B are equally distant from each other at their extremities ; 
hence, having their lengths always in the same direction between 
their extremities, they are, therefore equally distant from each 
other at every part between their extremities, each to each. 
D E and C B being straight lines, can be produced to any distance 
on either side, and still have their lengths in the same direction, 
consequently they will always be the distance D C or E B from 
each other. 

I have omitted the necessary references in these notes ; but 
in Benson's Geoimetry they will be found in their proper places, 
and my demonstrations fully sustained by them. 

proposition xvi. 

By means of the fifteenth proposition, it is easy to prove, with- 
out the Reductio ad Absurdum, this proposition and its converse, 
which is to show that token a straight line falls upon two parallel 
straight lines, it makes the alternate angles equal to one another, the 
exterior angle equal to the remote interior angle on the same side, 
and the two interior angles on the same side, together equivalent to 
two right angles. Other geometers always use the Reductio ad 
Absurdum to demonstrate this proposition. 

PROPOSITION XXIV. 

This is the general proposition for the equivalence of paral- 
lelograms described on the sides of any triangle ; and a particular 
case is that the square on the hypothenuse of a right angled tri- 
angle is equivalent to the sum of the squares described on the 
two sides about the right angle. In this proposition various 
methods are given to demonstrate the particular case. And all 
through Benson's Geometry various methods are either given, 



1 6 CONCERNING THE CIRCLE. 

pointed out or suggested, to solve the important geometrical 
propositions. All the demonstrations are conducted by the direct 
reasoning substituting entirely the Reductio ad Absurdum. In 
Benson's Geometry the whole science is condensed in six books, 
thus enabling the pupil to gain quickly a practical knowledge of 
this fundamental branch of Mathematics, and its various applica- 
tions to Trigonometry, Surveying, Mechanics, Engineering, Naviga- 
tion and Astronomy. Numerous and copious corollaries and 
scholiums are given all through the work, and important 
exercises are also appended, thus making Benson's Geometry, 
either for reference or text-book, the most complete work ever 

PUBLISHED. 



CONCERNING THE CIRCLE. 



No subject has ever engaged so much attention from profound 
thinkers as that of the Quadrature of the Circle, or what is popu- 
larly called "Squaring the Circle." From the earliest accounts of 
mathematical science, geometers and others have given the most 
assiduous application to discover its solution, with the invariable 
result of ill-success and despair ; until finally mathematicians as- 
sume a kind of indifference, and pretend to ignore all attempts 
for it. 

The true inquiring mind, however, essays to overleap all ob- 
stacles, becomes keenly whetted by impediments, and regards 
nothing known until everything is discovered. 

The problem is simply to find some method by which the area of 
a circle can he definitely determined. There should be nothing im- 
possible or even difficult in doing so, if the correct reasoning be 
used. The area of any triangle, square, or polygon is easily 
determined : and why should that finite geometrical figure, called 
the circle, be impossible of having its area determined ? If math- 
ematicians would disabuse their minds from all prejudice, discard 
all notions engendered by paralogistic reasoning, and regard the 
problem in a calm and rational manner, they will find that the de- 
termining cf the area of the circle is a legitimate subject for ge- 
ometrical investigation, and its accomplishment the positive result 
of proper analytical research. 

The great bugbear which frightens timid investigators into the 
"Polar Region" of mathematical science is, that because no one 
has ever yet been able to discover the area of the circle, ergo, the 
area cannot be discovered. 

Now, has it never occurred to mathematicians to inquire the 
reason why all the attempts to determine the area of the circle 

B 



1 8 CONCERNING THE CIRCLE. 

have been futile ? Surely this is a question worthy of considera- 
tion, and if it be earnestly entertained, much will be done to re- 
move the stigma now resting on the otherwise fair escutcheon of 
mathematical science. It is correctly believed that mathematicians 
argue from indisputable premises, that their reasonings from these 
premises are unimpeachable aphorisms, and that their conclusions 
are the legitimate deductions from those aphorisms. . Hence, it is 
clear why so much credence and utility are attached to mathemati- 
cal truths. Hence, "mathematically correct" is synonymous with 
absolutely true. And in consequence of the numerous failures to 
determine the area of the circle, when mathemetical reasoning is 
so rigidly correct, it is incumbent upon every mathematical in- 
quirer to ascertain whether the arguments for the area of the circle 
have been based upon indisputable premises and whether the de- 
ductions from those premises have been legitimately made so as 
to give the desired conclusion. These are the questions which 
should be uppermost in the mind of every mathematician, and if 
it be clearly shown that the arguments for the solution of the 
problem have not been based upon correct data, or that the con- 
clusion of the arguments has not been derived by proper deduc- 
tions, then it behooves every inquiring mind to rectify the error : 
and by rigid adhesion to the tenets and principles cf mathemati- 
cal science proceed to the desired result. 

Years of unceasing labor, intense and earnest application and 
the most self denying patience have been devoted to the calcula- 
tion of the ratio between the diameter and circumference of the 
circle ; operations have been conducted with figures having over 
one hundred decimal places, yea, in some instances exceeding 
two hundred decimal places, thus exhibiting the most wonderful 
intellectual feats, mental endurance, and human patience in the 
history of man ; entire lives have been spent upon the calculations ; 
and it is startling to think that all have been in vain to discover the 
area of the circle. No wonder then that mathematicians regard 
the subject with aversion and despair, and are prone to ignore 
every attempt to discover the area of the circle. ' The question 
has already been propounded in this article, why should the circle 
be impossible of having its area determined? But one answer has 
ever been given to this question. Because no one has ever yet 
been able to discover the area of the circle, ergo, the area cannot 
be discovered. Is it possible that such profound and acute thinkers 
as mathematicians, can be content with this answer ? What would 
the extent of human knowledge be to-day if Plato, Aristotle, 
Socrates, Archimedes, Galileo, and other philosophers, had 
been content with the information prevailing in their respective 
days, and had argued that everything was known that could be 
known ? Then,indeed, would Darwin's notion be most unmistakably 



CONCERNING THE CIRCLE. " ["*'§' 

_____ \ 

confirmed. Men having to Apes returned mil stfrom Apes have sprang. | 
Whatever may have been the origin of Man, his career is a pro- 
gressive one, and in the language of Macauley, — his goal of to-day 
is his starting point tomorrow. Let us give a few moments con-! 
sideration to the immense calculations which have been made 
upon the ratio between the diameter and circumference of the 
circle. The nature of decimals is like the relation of Time to 
Eternity ; man's mind is so constituted that he cannot grasp the 1 
notion of infinity, and in using decimals he deals with quantities 
that to him appear exhaustless, but to his uses and purposes the 
passing of Time has all the characteristics of Eternity, and so 
likewise, the value of quantity is not impaired by expressing it 
with decimals, because decimals can be carried so far that their 
difference from their Integer can be so much reduced as to be 
scarcely distinguishable or appreciable. Although, however so 
many the number of decimal places there be, the Integer is not 
reached, still this fact does not impair the value of quantity ex-, 
pressed by decimals ; witness the square root of a prime number, : 
this square root can be expressed by decimals only, but the square' 
root is none the less correct by being so expressed ; hence, then, 
decimals will not of themselves impair the value of quantity. And 
for all purposes, when in the case where the number of decimal 
places exceed two hundred, and the calculation conducted thereon, 
the result of the calculation sufficiently denote the ratio between 
the diameter and circumference of the circle, and the correct ratio 
in that case is as much ascertained as can be the ratio between 
the diagonal and side of a square ; now the square of that diagonal 
is double the square of the side ; hence, although the latter ratio 
can be expressed by decimals only, still the result is "mathemati-f 
cally correct," or in other words, absolutely true. Therefore we; 
see that the use of decimals is within the province of mathematical 
truth ; consequently the difficulty with the problem of the area of 
the circle is not on account of the use of decimals in the calcula- 
tion ; but it arises from something else. Mathematicians use the 
ratio found as stated above, as an element in their calculation for 
the area of the circle. Just here we will investigate whether such 
procedure is correct. In using the ratio in this case, mathema- 
ticians adopt a method of reasoning for the circle analogous to; 
their method of reasoning for the polygon. This is certainly not 
correct because the circle cannot be proven to be a polygon. In 
the case of a polygon when its perimeter is determined and its 
apothem is known, the area is easily ascertained, but when the 
circle is not a polygon, knowing the length of its circumference 
and its radius will not avail for determining the area of the circle ; 
that is, before the circle is proven to be a polygon, we cannot 
say that the half product of the circumference and radius will be 



20 CONCERNING THE CIRCLE. 

the area of the circle. It is for the reason that mathematicians 
assume the above product to be the area of the circle that their 
attempts have been futile. Mathematicians have devoted them- 
selves to determine the ratio between the circumference and 
diameter, believing when they have determined the ratio, that 
they have then solved the problem of the circle; but, however 
exact the ratio be made, their attempts to discover the area of the 
circle will be futile in using the ratio as an element in their cal- 
culation before the circle is proven to be a polygon, or before the 
area of the circle is proven to be the half product of the circum- 
ference and radius. Neither is possible to be proven, hence 
mathematicians have been for ages arguing from a wrong datum, 
hence, the reason is plain why all their attempts to discover the 
area of the circle have been abortive. Not that the ratio between 
the circumference and diameter of the circle has not been suffi- 
ciently determined; but, because mathematicians have treated 
the circle as a polygon, which is inconsistent with geometrical 
truth, hence their reasoning for the area of the circle is fallacious. 

About thirteen years ago, I published a new demonstration to 
determine the exact area of the circle ; and shortly afterwards I 
offered a reward of one Thousand Dollars to any one who 
refuted my demonstration. My offer was accepted, and various 
discussions ensued with several mathematicians. Committees of 
expert mathematicians were appointed to adjudicate upon the 
issues involved ; but no decision was rendered nor award made 
by them. 

And when I was in London, England, during the year 1864, I 
submitted my views to Prof. G. B. Airy, Astronomer Royal of 
Great Britain, and to the Royal Society of London. General 
Edward Sabine, who was then President of the Royal Society, 
received my manuscript, and submitted it to Prof. Stokes, one of 
the Secretaries of the Society, whom he described as " a math- 
ematician of the first order, and better able than himself to 
consider such subjects." Both Profs. Airy and Stokes declined 
to venture an opinion. 

The several points of the discussions I have already published 
in a pamphlet, entitled " Replies by Prof. Lawrence S. Benson, 
Author of Benson's Geometry, to Prof. E. T. Quimby, Dart- 
mouth College, Hanover, N. H. ; Prof. William Chauvenet, 
LL.D., Washington University, St. Louis, Mo. Prof. Robert 
D. Allen, Kentucky Military Institute, Farmdale, Ky. ; and 
Prof. A. T. Bledsoe, LL.D., formerly University of Virginia, 
now Editor Southern Review, Baltimore, Md." 

I made issue with the Reductio ad Absurdum method of reason- 
ing, and I remodeled the Elements of Euclid so as to use the 



CONCERNING THE CIRCLE. 21 

direct mode of demonstration entirely, which change has received 
the approval of eminent mathematicians everywhere : at the same 
time, no revolution in Thought can be accomplished without some 
clog to its wheel of progress ; and, however clear to some minds 
a principle may be, there are always to be found others who do 
not reconcile .themselves readily to what they regard as innova- 
tions, hence, from remote antiquity to the present time it is easy 
to cite innumerable instances, where things which are now consid 
ered true and unimpeachable, were obstinately opposed and 
denied, when first made known ; but opposition to Truth is always 
followed by beneficial results, and the struggle of Error with Truth 
has no doubtful termination, no matter how severe the contest ; and 
it is gratifying to know that when Truth conquers at last, Truth is 
established for evermore. 

There is no class of thinkers so tenacious of their opinion as 
mathematicians, for their opinion rests upon conviction, whereas 
other thinkers base theirs on belief. A geometrical demonstration 
is the highest evidence of truth that the human mind can entertain, 
and when a geometer follows a train of logical arguments from 
premiss to conclusion, the geometrical truth is so deeply implanted 
in his mind, that nothing can eradicate it. And, although I dis- 
sent from views which have been advocated since the dawn of 
Geometry, and supported by the illustrious Euclid and Archi- 
medes ; still, I am emboldened to appeal to the mathematician's 
proverbial Love of Truth and request a careful consideration of 
my convictions. 

In Arithmetic we have zero which signifies no quantity — and 
every quantity, however small, has an appreciable value distinguish- 
able from zero or nothing ; hence, the continual decrease of any 
quantity by divisions and sub-divisions can never produce entire 
destruction of value — therefore, it is plainly evident that however 
far the divisions and sub-divisions be continued, quantity can 
never be reduced to nothing or zero. This fact is conformable 
to one of the most important laws of Nature, which is, that 
destruction is impossible ; take any substance we choose, we can 
decompose that substance into its ingredients ; and it maybe pos- 
sible to decompose the various ingredients into more elementary 
bodies, still nothing has been destroyed, for though the bodies be 
further resolved into gases, and the gases be reduced still more 
to ethers — it is still within the province of Nature to re-compose 
those ethers to gases, the gases to various compound bodies, and 
finally reproduce the substance into its original entirety ; hence, 
destruction is incompatible with the nature of things, therefore, 
however so far the decrement of quantity be carried, there is an 
impassable barrier to that quantity being ultimately reduced to 
nothing. Hence, zero cannot be regarded as the ultimatum of 



22 CONCERNING THE CIRCLE. 

quantity ; and however near quantity may approach by continual 
decrements to zero, there will always be an appreciable difference 
between the most infinitesimal quantity and zero. Consequently, 
the limit of quantity is not zero. 

The limit of a variable quantity is not then that which is 
extraneous to quantity, but it is some unchangeable quantity into 
which variable quantities ultimately tend, which can be illustrated 
by fractions and unity ; all fractions find their limit in unity, be- 
cause, unity is the aggregate of fractions ; both unity and fractions 
are quantities, hence, variable quantities have a limit, and that 
limit is the aggregate of changes to which variable quantities are 
subject. And it is also evident that the limit of variable quantities 
is of the nature of those quantities. That is, there must be com- 
patibility between variable quantities and their limit ; for this 
reason, there being no compatibility between zero and quantity, 
zero is not the limit of quantity. 

This is true of OX finite quantities ; it is however contended by 
some thinkers, that when certain quantities converge to equality in 
any finite time, and before the end of that time approach nearer to 
each other than by any assignable difference, they will ultimately 
become equal. This is in substance the Twelfth Axiom of Euclid, 
which gave rise to so much discussion among geometers, and 
was adopted by Newton for the first lemma in his Principia, 
Geometers justly contend that since its converse has been de- 
monstrated, Euclid's axiom should also be demonstrated before 
its truth is accepted. It is a demonstrated truth that a curve and 
its asymptote, though continually approaching each other, never 
meet, however, so far they be produced on the same plane : 
but — contend some thinkers — if the lines be produced to infinity, 
they will meet. Forsooth, what do these thinkers know about in- 
finity ? The very supposition is absurd, the idea thztfinite quantities 
can become infinite is preposterous. The term — infinity — is 
very convenient to conceal paralogisms of reasoning, and the use 
of the term is meaningless and useless. How can the finite mind 
embrace the notion of infinity ? Universe signifies everything 
known and unknown, beyond which there is nothing conceivable, 
hence, universe is a limit to all human conceptions*, therefore, the 
infinite has no meaning in the apprehension of man. Again, it 
will be admitted, that infinity means never-ending, then, if it be 
granted that the curve and its asymptote meet in infinity, and 
since infinity is never-ending, then the lines never meet, which con- 
firms the truth already stated ; so that using the term infinity does 
not prove that the lines ever meet. Euclid, however, used his 
axiom for the consideration of straight lines, that is, for quantities 
of the same nature, and the axiom will not apply to the curve 
and its asymptote or quantities of dissimilar natures, which was 



CONCERNING THE CIRCLE. 2 j 

made evident also in the case of quantity and zero. Hence, a 
sameness is required, in the nature of quantities, compared to- 
gether. It is contended that Newton did not apply his first 
lemma to a quantity before or after it vanished, but at the time it 
vanishes. Now, the clearest interpretation to this is., that New- 
ton made a distinction where there is no difference, for a vanish- 
ing quantity is necessarily previous to a vanished quantity, hence, 
in applying his first lemma to a vanishing quantity, he necessarily 
applied it to a quantity before it vanished therefore, his first lemma 
applies to quantities previous to their approach to equality, and 
consequently, his first lemma does not prove ultimate equality 
between quantities which approach each other, even though their 
difference is less than any assignable quantity. This lemma of 
Newton is regarded as " the cornerstone and foundation of his 
Prinajna ; " hence, its insufficiency sweeps away " the support of 
the entire system." This lemma of Newton is used to show that 
when polygons are described about the circle, and their sides are 
continually increased in number, that ultimately they meet and 
form the circumference of the circle ; but the lemma does not 
prove equality between quantities that continually approach each 
other in a. finite time, and it is assumed that when the time is infinite 
then the quantities become equal, which is tantamount to say : 
that the quantities never become equal, for infinite fime is never- 
ending time, hence, if the quantities become equal in never-ending 
time only, the quantities then never become equal. Again, it has 
been shown that there must be a sameness in the nature of quan- 
tities in order that there* can be a limit to their difference, quantities 
which are fundamentally different cannot reach equality, there- 
fore, it is impossible for straight lines, however, small they be 
reduced, to lose their characteristics and emerge into a curve, 
consequently, however small the sides of the polygons be reduced, 
those sides will always retain the characteristics of straight lines, 
and can never emerge into the curve ; and the notion that a. finite 
curve is composed of an infinite number of straight lines is 
contrary to human reason, as it is a contradiction of terms, for it 
would bring infinity within the bounds oifinity. 

When the geometer states that the circumference is a curve 
that is at every part equally distant from the centre of the circle, 
he simply expresses the condition that makes the equality of 
radii ; the radii are straight lines ; hence, he expresses the condi- 
tion that makes the constant equality of certain straight lines. 
When the geometer states that if two points be given in position 
on a plane, and another point be conceived to move about them, 
in such a manner that the difference between the distances of the. 
moving point from the stationary points be always constant, the 
moving point will describe the curve of the hyperbola, he has 



24 CONCERNING THE CIRCLE. 

simply expressed the condition that makes the difference between 
two varying straight lines always constant. When the geometei 
states that the periphery of an ellipse is such, that if from two 
given points on its transverse axis, two straight lines be drawn to 
any common point in the periphery, me sum of these two straight 
lines will be always equal to the transverse axis, he has simply ex- 
pressed the condition that makes the sum of two varying straight 
lines always constant. And when the geometer states that if a 
point and a straight line be given in position on the same plane, 
and another point be moved in such a manner that the distance 
of the moving point from the fixed point be always equal to the 
distance of the moving point from the fixed straight line, the 
moving point will describe a parabola, he has simply expressed 
the condition that makes the constant equality of two varying 
straight lines. Hence, the circumference of the circle shows how 
the equality of straight lines can be always constant. The 
hyperbolic curve shows how the difference between two varying 
straight lines can be always constant ; the elliptic curve shows 
how the sum of two varying straight lines can be always constant, 
and the parabolic curve shows how the equality of two varying 
straight lines can always be constant. Whilst to form the alge- 
braic equation of the circle, the geometer uses the relations be- 
tween straight lines, viz : the abscissa and ordinate of a point in 
the circumference, and from the same rectangular co-ordinates, he 
forms the polar equation of the circle, whilst the algebraic equa- 
tions of the hyperbola, ellipse and parabola are formed from the 
relation of certain straight lines drawn in or about those figures, 
and their polar equations are also derived from the system of 
rectangular co-ordinates. The transcendal equations of curves are 
similarly derived as the algebraic, but they are expressed dif- 
ferentially. The exponential equations of curves are interme- 
diately between the algebraic and transcendal equations, where one 
or both of the unknown quantities enter as exponents, and are 
the methods of performing algebraic operations on exponential 
quantities. Hence, all the equations of the circle, ellipse, hyper- 
bola, parabola, curves of double curvature, and curved surfaces 
are formed from the properties of straight lines. Since elementary, 
analytical, descriptive and perspective Geometry, the algebraic 
Analysis, the differential and integral Calculus are based upon the 
principles of Elementary Geometry ; and since these principles in 
no wise comprehend the characteristic properties of the curve, 
these properties have been excluded from mathematical consider- 
ation. v . 

The application of the Differential and Integral Calculus to 
curvilinear magnitudes is based upon the notion that the finite 
curve is an infinite number of straight lines, hence, its fundamental 



CONCERNING THE CIRCLE. 



5 



is a contradiction of terms, and consequently, the Calculus has the 
taint of error throughout every operation concerning curvilinear 
areas. Hence, the infinitesimals of Calculus are entirely the 
infinitesimals of rectilinear magnitudes ; and Calculus is absolutely 
a system of rectilinear truths. The process of mensuration for 
rectilinear areas is based upon the property of Parallel Straight 
Lines that they are equally distant from each other, however so 
far they may be prolonged on the same plane. From this 
property the contents of the rectangle is derived ; now, any rect- 
angle is resolvable into two equal triangles, therefore any triangle 
is half the rectangle on its base and altitude, and, since all recti- 
linear surfaces can be resolved into triangles, the contents of all 
rectilinear surfaces are consequently derived by the above property 
of Parallel Straight Lines. 

But this property of Parallel Straight Lines has no application 
to Curves, and no curvilinear area can be resolved into triangles ; 
hence, the process of mensuration for rectilinear areas cannot be 
applied to curvilinear areas, therefore, these latter areas require 
a process of mensuration compatible with their peculiar nature 
and properties ; for. tin s reason when the rectilinear process of 
mensuration is applied to curvilinear areas, approximative results 
only are obtained ; but Geometry is an exact Science, therefore, 
the approximative results are not consistent with the strictness of 
geometrical reasoning. 

In the application of the rectilinear process of mensuration to 
curvilinear areas, the curve line is assumed to be an infinite 
number of straight lines ; a contradiction of terms, a contradiction 
of the definitions given by geometers for the straight and curve 
lines, and a contradiction of an established geometrical truth — for 
it would make circumscribed and inscribed polygons equal to the 
circle when their perimeters are equal to the circumference, when, 
in fact, the circle contains more area than any other figure of equal 
perimeter. 

And when the rectilinear process is applied to polygons of a 
great number of sides inscribed in the Parabola, an excess is 
obtained for their areas over the area of the Parabola, which 
shows most conclusively that this process of mensuration is 
unreliable for curvilinear areas. 

The application of this process to curvilinear areas cannot be 
made tenable by valid arguments; but is attempted to be sustained 
by what is called the Reductio ad Absurdum, a species of sophistry 
whose arguments are based upon suppositions known to be false, 
for instance, to prove equality between two magnitudes, a 
circle A, and a rectangle B, described on the radius of circle 
and a straight line equal in length to the semicircumfer- 
ence. "For the sake of argument," the rectangle is sup- 



26 CONCERNING THE CIRCLE. 

posed equal to another circle C, greater than the rectangle, and 
because, absurdity results, it is concluded that the rectangle can 
not be equal to any circle greater than the circle A, when, in the 
process of reasoning, the properties of the circle A were entirely 
ignored, and the argument conducted altogether upon the rela- 
tions between the rectangle and the circle C ; therefore, the rect- 
angle is not shown equal to the circle A, because the circle C is 
not prove?i to exceed equally circle A and the rectangle. And in the 
supposition that the rectangle is equal to a circle D, less than the 
rectangle ; because absurdity results again, it is concluded that 
the rectangle cannot be equal to any circle less than circle A, 
when, in fact no proof is given that the circle D is exceeded 
equally by the rectangle and circle A, which must be shown in 
order to prove equality between the rectangle and circle A. 
Hence, the application of the rectilinear process of mensuration 
to curvilinear areas, being untenable, it becomes necessary to use 
a process of mensuration for curvilinear areas more in accordance 
with them. 

For instance, the contents of a rectangle is found by multiply- 
ing its base and altitude together; but the contents of a trapezoid 
is not found by multiplying its base and altitude together, although 
they are both rectilinear figures, then how much more necessary 
is it to use different processes of mensuration for a polygon and a 
circle when they are not both rectilinear figures. 

The circle is a surface of revolution, being produced by revolv- 
ing in the same plane a straight line around one of its extremities, 
which remains fixed, therefore, the properties of the circle are 
made known by the process of revolution ; it was by revolution 
that Archimedes discovered the relations and properties of the 
solidities and convex surfaces of the Cone, Sphere and Cylinder, 
hence, the process of revolution is of long established use, and 
when it is used in connection with the Axiom that "if two 
equivalent surfaces be moved in a line perpendicular to their plane, 
or revolved, the one which moves farther generates the greater 
volume" — the conclusion thereby derived is in accordance with 
the peculiar properties of curvilinear magnitudes; hence, this 
conclusion differs from the approximative result derived by the ap- 
lication of the rectilinear process of mensuration to curvilinear 
areas, which should necessarily be so. The rectangle contains 
more area than any curvilinear surface on the same base and 
altitude, the rectilinear perimeter is greater than the curvi- 
linear perimeter wizen the areas are equal, and in the generation 
of equivalent volumes, by revolving equivalent rectilinear and 
curvilinear sections around the same axis, with same radius and 
altitude, the perimeter of the rectilinear section is greater than 
the perimeter of the curvilinear section. Hence, since it is 



CONCERNING THE CIRCLE. 



27 



from their bounding lines that the properties of geometrical figures 
are derived, it is evident, that the properties of rectilinear areas 
are necessarily excessive of the properties of the curvilinear areas, 
which has been actually demonstrated in the case of inscribed 
polygons in the Parabola. 

The above disagreements show that the different processes are 
in accordance with the respective properties of the rectilinear and 
curvilinear areas ; the fact that the excess is by the rectilinear pro- 
cess shows that the process of revolution conforms to the peculiar 
properties of curvilinear areas, and the fact that the rectilinear 
process gives approximative results when applied to curvilinear 
areas shows that it is not consistent with the strictness of geomet- 
rical reasoning to use the rectilinear process to determine curvi- 
linear areas. 

The Method of Revolution gives the Exact area oj the Circle* 
A B 




C D 

In the above diagram the square A B C D revolved around 
either A C or B D as an axis, will generate a cylinder. 

B D C E revolved around the axis B D will generate a volume 
equivalent to two-thirds of the cylinder generated by the square 
A B C D. 

BCD revolved around the axis B D will generate a volume 
equivalent to one-third of the above cylinder, hence, B E C re- 
volved around the axis B D will generate a volume equivalent to 
one third of the same cylinder, hence, also B A E revolved around 
the axis B D will generate a volume equivalent to one-third of 
the same cylinder ; therefore the line B E divides equally the 
volume generated by B A C around the axis B D, and the line 
B E divides equally the triangle BAG. Hence, BECF revolved 
around the axis B D generates a volume which is equivalent to 
one-half of the cylinder generated by the square A B C D. 

Now when the arcs B G C and BHC are described with radii 
A B and C D, the volume generated by B G C D around the 
axis B D is equivalent to two-thirds of the cylinder generated by 
the square A B C D, hence the segment BGC revolved around 



28 CONCERNING THE CIRCLE. 

the axis B D generates a volume equivalent to one-third of the 
above cylinder, therefore, the arc B G C divides equally the vol- 
ume generated by B A C around the axis B D. In the same 
manner, it can be shown that the arc B H C divides equally the 
volume generated by B D C around the axis A C. Since BCD 
revolved around B D is equivalent to B C A revolved around 
A C ; and since the volume generated by B C D around the 
axis B D, is one-half the volume generated by B A C around 
the axis B D, and the volume generated by B A C around the 
axis A C is one-half the volume generated by BC D around the 
axis A C, these volumes then have reciprocal relations whether 
around one axis or the other. Hence, the volumes are equally 
divided by the straight lines B E and C F, and by the arcs B G C 
and B H C. Therefore, the volume generated by B G C H around 
either of the axes is equivalent to one-half of the cylinder generated 
by the square A B C D. 

Since the volumes generated by B A C, and B D C either around 
the axis B D or the axis A C, are equally divided by the straight 
lines BE and C F; and as the triangles B A C and BDC are also 
equally divided by the same straight lines ; and since the same 
volumes are equally divided by the arcs B G C and B H C, so also 
the same arcs equally divide the triangles B A C and BDC. 

Consequently, BEC F and B G C H generate equivalent 
volumes, whether around the axis B D or the axis A C ; also the 
volumes generated by B A C G and the segment B G C, are 
equivalent ; also, the volumes generated by B E A and the segment 
B G C are equivalent, whether around the axis B D or the axis 
A C ; hence, these surfaces generating equivalent volumes under 
the same conditions, the surfaces are prima facie, equivalent to 
each other. Therefore, B G C is one-four thoi the square A B C D ; 
the quadrant B G C D is three-fourths of the square A B C D, 
and the circle is three-fourths of the square of its diameter, or, the 
arithmetical mean between the circumscribed and inscribed squares. 
Again : by the process of revolution, a square generates a cyl- 
inder ; a triangle one-half the square, and having equal base and 
altitude with the square, generates a cone, which is one-third of 
the cylinder ; and a trapezoid, three-fourths of the square, and 
having equal base and altitude with the square, generates a volume 
which is two- thirds of the cylinder ; hence, a mean surface between 
the square and triangle having equal base and altitude with the 
square, generates a mean volume between the cylinder and cone. 
A quadrant of circle with a side of square as radius, generates a 
hemi-sphere — a mean volume between the cylinder and cone, con- 
sequently (Ax. i, Bk. i, Benson's Geometry) the quadrant is a 
mean surface between the square and triangle. 

The results obtained by revolution perfectly agree with estab- 



CONCERNING THE CIRCLE. 29 

lished geometrical truths relating to curve surfaces. Archimedes, 
by revolving surfaces around an axis, discovered that the cone, 
sphere and cylinder are to one another as 1, 2, 3 ; and the same 
procedure gives the proportion between the solidities ana surfaces 
of the sphere and cylinder as 2 to 3 ; it makes the convex surface 
of the cone equivalent to one of the great circles of the sphere ; 
the surface of the sphere, four times the area of one of its great 
circles, and the convex surface of the cylinder, equivalent to the 
surface of the sphere, and it makes the solidity of the cone one- 
third of the cylinder, and the solidity of the sphere two-thirds of 
the cylinder. These results being obtained, by the method of 
revolution, we have consequently an agreement with them of the 
area for the circle, obtained by revolution ; for instance, the area 
of the circle being proved by this method equivalent to three- 
fourths of the square of its diameter, we get the proportion of 
1, 2, 3, 4, for the cone, sphere, cylinder and cube, thus agreeing 
with Archimedes' proportion.* The sphere being one-half of the 
cube, we have its surface equivalent to four times the base of the 
cylinder, equivalent to four times one of its great circles, or two- 
thirds of the surface of the cylinder. The cone being one-fourth 
of the cube, we have its convex surface equivalent to one of the 
great circles of the sphere, and the cylinder being three-fourths of 
the cube, we have its convex surface equivalent to the surface of 
the sphere. Thus we see the results of revolution possess the 
chief characteristic of geometrical theorems, for they have a per- 
fect congruity with one another, and preserve a consistence which 
renders them useful and reliable. " Facts are stubborn things." 

Very respectfully, 

LAWRENCE S. BENSON, 

Author of " Benson's Geometry.'' 
New York City, September, 1873. 

* See Benson's Geometry, pages 166, 216, 217. 



ADVERTISEMENT. 



Benson's Geometry contains the Elements of Euclid and Ee- 
gendre — simplified and arranged to exclude the illogical Re- 
ductio ad Absurdum — a full and explicit treatise on Plane and 
Spherical Trigonometry ; also, interesting and valuable exercises 
in Elementary Geometry and Trigonometry. 

The endorsements it has received from prominent mathema- 
ticians in all parts of the country, and the close scrutiny and 
critical examinations given to it by other prominent mathema- 
ticians, through which it has successfully passed, entitle Benson's 
Geometry to be esteemed as the most correct, the most 
progressive and the most complete Geometry that has ever 
been published. 
• The success of Prof. Benson's views is shown in their practical 
endorsement by President Hunter, of the Normal College of 
the city of New York, and by the recent repudiation of the Re- 
ductio ad Absurdum by Prof Charles Davies, LE.D. 

Benson's Geometry is adapted as a Text Book for Schools and 
Colleges ; it is elegantly bound in sheep, and has already passed 
through several editions. Price $2.00 per copy. A liberal dis- 
count to schools and colleges. 

l€zT All persons wishing Benson's Geometry must enclose 
$2.00, which includes postage. No attention given " to letters 
without the money. 



TESTIMONIALS. 

From Trof. Jas. &. Carlisle, TTofford College, S. C. 

" Some of your definitions strike me as improvements on those which have 
been copied from one work into another, for many years. Tour judicious 
selections from English works not in common use in our country, and your 
own contributions to this old, but still fruitful science, will give a peculiar 
value to your work." 

■ . 

From 'Prof. Harris, Zexington, Georgia, 

" I have carefully examined your 'Elements of Euclid and Legendre,' and 
pronounce it the best work of the kind I have seen. I am pleased with your 
direct course of reasoning, and general classification of the work. Tou have 
succeeded admirably in combining the essential propositions of geometrical 
science in so small a compass, that, in my opinion, a pupil may learn, yes, 
master the whole science, in the same time that he, in former text books, 
would have completed five books of it. Tour exclusion of the l reductio ad 
absurdwn ' is a decided improvement in geometrical science." 



From Prof. J~. *D. Stewart, Classical High School, 
Memphis, Tenn. 

"I have just read your circular addressed to teachers in the United States. 

I am at once convinced of the justness of the points you make against the 
Legendre of Davies. I hail with delight the abandonment of the 'redudio 
ad absurdwn ' method of reasoning. Allow me to congratulate you on this 
advance in the science of Geometry — a science which is the foundation of 
the mathematics." 



From, the Scie?itific American. 

"The Elements of Euclid and Legendre, with Elements of Plane and Spherical Trigonometry. 
By Lawrence S. Benson. 

" The author of this treatise has prepared and published a text-book, adap- 
ted for the use of schools and colleges, the plan of which being the reducing 
of geometrical science to the smallest compass, such propositions are only 
introduced in it, as are required to substantiate the principal theorems, by 
which the principles of geometry have practical applications in trigonometry, 
surveying, mechanics, engineering, navigation and astronomy. A new and 
important feature of this work, is the establishment of all geometrical propo- 
sitions by the direct method of reasoning, dispensing entirely with the re- 
dadio ad absurdum, or indirect demonstration ; the author's argument 
being that every true proposition must be susceptible of proof without any 
circuitous process, as that heretofore employed for demonstrating certain 
propositions. The work before us bears the commendation of President 
"Webster and Professor Docharty of the College of the City of Xew York, 
Professor J. GT, Fox, Principal of the Cooper Union Free Schools, also the Su- 
perintendent of the Board of Education, of this city, and has been entered on 
the list of text-books for the Ward Schools of this city," 



ytz vi#iT ^o ¥8s( 0tf>r : 



OR 



Critical Essays on Physics, Metaphysics and Ethics, 



BY 



PROF. LAWRENCE S. BENSON, 
Author of BENSON'S GEOMETRY. 

Revised and Enlarged from the English Edition. 

IN PREPARATION FOR THE PRESS. 



Prof. G. B. Dochartt, LL.D., of the College of the City of New York, 
writes : 

" I have had several interviews with Mr. Lawrence S. Benson, on scientific 
subjects, and from his conversation, together with the essays which he has 
published, I esteem him an excellent scholar and fine mathematician." 

" The views of the author are presented in the form of an allegory. He 
conceives himself on the Sun, where he surveys the universe ; he pictures to 
himself the aspects, relations and conditions of things totally different from 
what they seem on the Earth ; and from his exalted standpoint he criticises 
the theories of Gravitation, Central Forces, Projectiles, Hypothetical Astron- 
omy, Atomic Chemistry, Electricity, Galvanism and Magnetism ; he advances 
some new notions of Matter, and proclaims the seeming paradox that the 
Heat we feel in the sunlight is not emitted to us from the sun, and that we 
never can obtain any knowledge of Cause. He promises, at some near future 
time, to dissipate the mists of Metaphysics and to solve some Knotty ques- 
tions in Ethics. The work is original in conception, comprehensive in plan, 
bold in character, vigorous in style, exhaustive in research, forcible in argu- 
ment, poetical in diction, and erudite in treatment* It challenges the scrutiny 
of savans ; it is readable and instructive, and will repay many perusals. We 
are anxious to see the continuance of it, as we regard it as a wonderful 
work, and we esteem it as a valuable acquisition to our stock of scientific 
lore." — Reviewer. 

Address JAMES S. BURNTON, Publisher, 

149 Grand Street, New York. 






: Jeplg to ^riticomji on neon's |)eotm|iri>. 



-r 



)t\he recent edition of " Notes on the First Book of Benson's 
-L Geometry," &c, has brought out some remarks from Prof. 
F. O. Marsh of Denison University, Granville, O., from Prof. 
James Curley of Georgetown College, District of Columbia, and 
from Prof. E. T. Quimby of Dartmouth College, Hanover, New 
Hampshire. The last two contend as follows : That when equiv- 
alent volumes are generated by surfaces whose points describe une- 
qual circles around the common axis, the surface whose points 
describe the greater circles is less than the surface whose points de- 
scribe the smaller circles. Whence Prof. Curley instanced BGC 
and B G C A when B D is the axis, because the circles formed 
by the points of B G C A are greater than those formed by the 
points of B G C ; and since these surfaces generate equivalent 
volumes around B D, he reasons that B G C A is less than BGC. 
And Prof. Quimby argues that, for similar reasons, B G is greater 
than G E C. 



B 



D 




A E C 

In reply, 1 called their attention to the triangles B E C and 
B F C. They are equal, and they generate equivalent volumes 
around B A ; but the circles formed by the points of B F C are 
larger than those formed by the points of B E C. And when 
BECF and B G C H are revolved around D C, they generate 
the same volumes that they do around B D, but when D C is the 
axis, B G forms larger circles than G E C forms, and when B D is. 



/S7X 



2 REPLY TO CRITICISMS. 

the axis, B G forms smaller circles than G E C forms ; now, if B G 
be greater . than G E C, when D C is the axis, B G C H and 
BECF should not form the same volumes that they do when 
B D is the axis. And it will again be seen that when B G C H 
and BECF are revolved around B D, the points of B E C F 
form larger circles than the points of B G C H form ; and when 
they are revolved around A B, the points of B E C F form small- 
er circles than the points of B G C H form ; nevertheless, these 
surfaces generate the same volumes around either axis. Hence 
the relation of B E C F and B G C H is that of equivalence ; 
consequently, B G C and B G C A are equivalent — so, also, are 
B G and G E C, and the circle is 3R 2 . 

Profs. Curley and Quimby contend, also, That equivalent sur- 
faces must have their centres of gravity equally distant from the 
axis in order to generate equivalent volumes. 

Prof. Curley, therefore, contends because the centre of gravity 
of B G C A is farther from the axis B D than the centre of gravity 
of B G C, that B G C is greater than BGCA; and Professor 
Quimby, for similar reasons, contends that B G is greater than 
GE C. 

To this, I reply that BECF andB G C H have a common cen- 
ter of gravity, and since B C H bears the same relation to B C D 
that B G C bears to B A C ; and B F C bears the same relation 
to B C D that B E C bears toBAC; and because B F C and 
B H C revolved around A C generate the same or equal volumes 
that B E C and B G C generate around B D, and vice versa, that 
therefore, the conditions in the rotation of B G C and B H C are 
the same as the conditions are in the rotation of B E C and B F C. 
Since B F C generates around B D one-half the volume generated 
by B E C around B D, it follows that BHC generates around 
B D one-half the volume generated by B G C around B D. Hence 
B G C H generates an equivalent volume with BECF. But 
B G C and B E C have not the same centre of gravity, neither do 
BHC and B F C have the same centre of gravity. Hence it is 
evident that it is not requisite for the points of the equivalent sur- 
faces alzoays to form equal circles in order that the surfaces gene- 
rate equivalent volumes ; neither is it requisite for their centers of 
gravity to be always at equal distances from the axis in order that 
the surfaces generate equivalent volumes. 

Prof. F. O. Marsh considers a fatal defect in my argument to 
be, where I reason Because the arc B G C equally divides B A C 
when revolved around B D, therefore B G C also equally divides 
B A C when revolved around AC I sustain this by citing the 
reasoning of Archimedes upon the relation between the cone, 
sphere and cylinder. He shows the relation of those volumes 



REPLY TO CRITICISMS. 3 

from the relation of their sections ; and since the reasoning of 
Archimedes has been accepted by all geometers, its converse is 
also true, that is, we can derive the relation between the sections 
of volumes of rotation from the relation between the volumes 
themselves. And since the same relation exists between the vol- 
umes generated by B D C and B F C around B D that there is 
between the volumes generated by B A C and B E C around B D 
it follows that the same relation exists between the volumes gen- 
erated by B D C and B H C around B D that there is between 
the volumes generated by B A C and B G C around B D. Be- 
cause the relation between B A C and B G C is the same when 
they are revolved around B D that it is when they are revolved 
around A C, and the conditions of their rotation are the same 
around one axis that they are around the other axis ; hence when 
B A C generates around A C one-half the volume it generates 
around B D, it follows that B G C will generate around A C one- 
half the volume it generates around B D. 

Hence it is seen that neither of the above issues made by Profs. 
Curley, Marsh and Quimby disproves that the circle is 3R 2 . 
They however make another issue to the effect, That when the 
areas of polygons inscribed in the circle are computed by means of 
plane triangles , a result is obtained for the inscribed polygons great- 
er than 3R 2 , and they reason That it, is impossible for a circle to 
be less than a figure inscribed in the circle. They dwell upon this 
point to prove the circle not 3R 2 , though unable to sustain them- 
selves on the other issues. 

Perhaps some explanations are due from me, showing why I 
hold the circle to be exactly 3R 2 , when by reasoning from plane 
triangles it can be shown that 3R 2 is the area of a dodecagon in- 
scribed in the circle, which seems to infer That a part can be 
equal to the whole. This interpretation of my reasoning is no 
doubt entertained by a great many mathematicians, and to such I 
will say that, for the past fourteen years, I have given much reflec- 
tion and research to mathematical subjects, and I trust that I have 
not committed so egregious a blunder as to bring myself in di- 
rect contradiction to one of the most self-evident and most gen- 
erally known propositions in Geometry. 

I am touching subjects held sacred by mathematicians, inas- 
much as they have been consecrated by long reverence, the ripe 
thought and the earnest convictions of the most illustrious and expe- 
rienced thinkers known in the history of man : but with all due 
respect for the treasured Wisdom of ages, gathered, condensed 
and utilized by great patience and indefatigable industry, I must 
say, that it will amount to nought if it shrinks from the wand of 
Truth : or if, from this pretext or that prejudice, it avoids the light 
of Inquiry. 



4 REPLY TO CRITICISMS. 

Torelli, the celebrated Italian geometer and learned com- 
mentator of the Writings of Archimedes, contends that if we are 
to reason from the relation which certain rectilinear figures be- 
longing to the circle have to one another, notwithstanding that 
those figures may approach so near to the circular spaces within 
which they are inscribed, as not to differ from them by any assign- 
able magnitude, that the circle will be proved to be to the square 
on its diameter exactly as 3 to 4. 

Playfair, who took notice of these circumstances without "any 
design to lessen the reputation of the learned Italian, who has in 
so many respects deserved well of the mathematical sciences," 
complains that a very gross paralogism is to be found in that* part 
of Torelli's reasoning, where he makes "a transition from the 
ratios of the small rectangles inscribed in the circular spaces, to 
the ratios of the sums of those rectangles, or of the whole rectilin- 
ear figures," and demonstrates that this proposition of Torelli is 
true on two conditions. (See Playfair' s Euclid, Sup. Notes, p. 307.) 
Now, the fact that Torelli's proposition is true on "two condi- 
tions " prevents the proposition from being false — for a false pro- 
position can be true on no condition. Hence Torelli, having 
reasoned from a proposition true on " two conditions," his conclu- 
sion has validity ; inasmuch as no result obtained by reasoning is 
absolutely true per se. Ev # ery Science, whether mathematical, 
metaphysical or physical, is simply a collection of those facts or 
deductions derived from certain fixed and limited conditions, which 
agree with and support one another. And it is impossible to rea- 
son on geometrical subjects without observing certain conditions : 
we say that a line is length without width, but the condition to be 
observed here is that a line must be regarded in the abstract, 
apart from any real or material condition ; hence, to call a line 
length without width is not absolutely true, per se, but simply, ab- 
solutely true with regard to certain conditions. Hence, Play- 
fair's objection to Torelli's proposition, loses its force from the 
very nature of geometrical truths. 

Playfair moreover objects that the conditions which make 
Torelli's proposition true " do not belong to the magnitudes to 
which he applies it." Now, Torelli applied his proposition to 
the relation of " the ratios of the small rectangles inscribed in the 
circular spac*es, to the ratios of the sums of those rectangles, or of 
the whole rectilinear figures." It is evident that when, the small 
rectangles are to each other in certain ratios, the sums of tlwse rect- 
angles are also in the same ratios. Hence, Torelli in making "a 
transition from the ratios of the small rectangles inscribed in the 
circular spaces, to the ratios of the sums of those rectangles or of 
the whole rectilinear figures," violated no principle of geometrical 
reasoning, and his conclusion is, therefore valid and sound. 



REPLY TO CRITICISMS. 5; 

Torelli having " studied the writings of Archimedes with un- 
common diligence," and " whose labours have done so much to 
elucidate the writings of the Greek Geometer," realized the fact 
that the circle is to the square on its diameter exactly as 3 to 4, 
which result is confirmed by the rotation of surfaces around an 
axis. 

When we make a transition from the ratios of the small rectan- 
gles inscribed within the Parabola, to the ratios of the sums of 
those rectangles or of the whole rectilinear figures, we prove the 
Parabola to be to the rectangle on its abscissa and ordinate exactly 
as 2 to 3. 

But when we compute the areas of inscribed polygons in the 
Parabola by means of plane triangles, we derive a result for the 
inscribed polygons in excess of that obtained for the Parabola. 
Now this is a parallel case with the circle and inscribed polygons, 
and it is evident that the similarity of the results can not be from 
accident. 

The question naturally arises, How is it possible for any result 
obtained by reasoning consistently with the principles of Geome- 
try to conflict with another result also obtained by reasoning con- 
sistently with the principles of Geometry ? 

In Geometrical demonstrations, the straight line and curve are 
used : the curve varies the direction of its length continually, 
whereas the straight line has its length always in the same direc- 
tion ; hence, the straight line is essentially and fundamentally dif- 
ferent from the curve ; so much so, that reasoning based upon the 
properties of the curve cannot agree with the reasoning based 
upon the properties of the straight line. Therefore, when reason- 
ing by means of plane triangles, which depends upon the proper- 
ties of the straight line, we obtain results which must necessarily 
differ from the results obtained by the reasoning by means of ro- 
tation which produces the curve forming the circle. Hence, it can 
at once be seen that the essential and fundamental difference be- 
tween the properties of the straight line and curve is a barrier to 
the properties of the respective lines being treated together under 
the same course of reasoning. 

The mode of reasoning adopted in Arithmetic affects numbers 
only in all their applications ; but cannot be used to falsify Alge- 
braic results arising from signs and quantities. For, always in 
Arithmetic we have an increase in the addition of numbers, and a 
diminution in the subtraction of numbers ; while from the mode 
of reasoning adopted in Algebra, a diminution is often obtained in 
the addition of quantities, and an increase in the subtraction of 
quantities. And would it be reasonable to argue that the results 
of Arithmetic and Algebra conflict with mathematical accuracy ?." 



6 REPLY TO CRITICISMS. 

No, because these results have relation merely to those matters 
which belong to Arithmetic and Algebra respectively. And it is 
the same with the results obtained by reasoning upon the proper- 
ties of the straight line and the curve ; they are true in their re- 
spective spheres, and their disagreement does not conflict with the 
strictness of geometrical reasoning ; it rather conforms to the 
strictest logical requirements. And it may be inquired, how is it 
that reasoning from plane triangles for the computation of the 
areas of polygons and reasoning from the ratios of rectangles, 
when they are all rectilinear magnitudes, that different and con- 
flicting results are obtained ? The answer is that the reasoning 
•on the ratios and rotation of surfaces involves their relation to each 
other : whereas the computation of the plane triangles involves 
their boundaries. And since for the Quadrature of the Circle 
the relation between the circle and a certain rectangular space is 
required ; it is evident that the proper mode of reasoning is by 
means of the relation of " the ratios of the small rectangles in- 
scribed in the circular spaces, to the ratios of the sums of those 
rectangles, or of the whole rectilinear figures ;" or, by means of the 
rotation of rectilinear and curvilinear surfaces around a common 
axis — and not by the process of continually doubling the number 
•of sides of the polygons described about the circle ; since the 
sides do not reach the circumference, this process gives an approx- 
imate result only which is inconsistent with the strictness of geo- 
metrical reasoning. 

Very respectfully, 

LAWRENCE S. BENSON, 

Author of "Benson's Geometry." 

New York City, November, 1873. 



TESTIMONIALS. 

From Trof. Jas. H. Carlisle, Wofford College, S. C. 

14 Some of your definitions strike me as improvements on those which have 
fceen copied from one work into another, for many years. Your judicious 
selections from English works not in common use in our country, and your 
own contributions to this old, but still fruitful science, will give a peculiar 
value to your work." 

From 1Prof. Harris, Lexington, Georgia. 

" I have carefully examined your 'Elements of Euclid and Legendre,' and 
pronounce it the best work of the kind I have seen. I am pleased with your 
direct course of reasoning, and general classification of the work. You have 
succeeded admirably in combining the essential propositions of geometrical 
science in so small a compass, that, in my opinion, a pupil may learn, yes, 
master the whole science, in the same time that he, in former text books, 
would have completed five books of it. Your exclusion of the ^reductio ad 
absurdum ' is a decided improvement in geometrical science." 



From iProf. «/". ID. Stewart, Classical High School, 
Memphis, Tenn. 

" I have just read your circular addressed to teachers in the United States. 
I am at once convinced of the justness of the points you make against the 
Legendre of Davies. I hail with delight the abandonment of the l reductio 
ad absurdum ' method of reasoning. Allow me to congratulate you on this 
advance in the science of Geometry — a science which is the foundation of 
the mathematics." 



From the Scientific American* 

"The Elements of Euclid and Legendre, with Elements of Plane and Spherical Trigonometry. 
By Lawrence S. Benson. » 

" The author of this treatise has prepared and published a text-book, adap- 
ted for the use of schools and colleges, the plan of which being the reducing 
of geometrical science to the smallest compass, such propositions are only 
introduced in it, as are required to substantiate the principal theorems, by 
which the principles of geometry have practical applications in trigonometry, 
surveying, mechanics, engineering, navigation and astronomy. A new and 
important feature of this work, is the establishment of all geometrical propo- 
sitions by the direct method of reasoning, dispensing entirely with the re- 
ductio ad absurdum, or indirect demonstration ; the author's argument 
being that every true proposition must be susceptible of proof without any 
circuitous process, as that heretofore employed for demonstrating certain 
propositions. The work before us bears the commendation of President 
"Webster and Professor Docharty of the College of the City of New York, 
Professor J. G. Fox, Principal of the Cooper Union Free Schools, also the Su- 
perintendent of the Board of Education, of this city, and has been entered on 
the list of text-books for the Ward Schools of this city." 



JV1T VI0IT TO TSSJ 0l/]Sf : 



OR 



Critical Essays on Physics, Metaphysics and Ethics, 



BY 



PROF. LAWRENCE S. BENSON, 
Author of BENSON'S GEOMETRY. 

Revised and 1 Enlarged from the English Edition. 



IN PREPARATION FOR THE PRESS. 



Prof. G. B. Dochartv, LL. D., of the College of the City of New York ; 

writes : 

" I have had several i^erviews with Mr. Lawrence S. Benson, on scientific- 
subjects, and from his conversation, together with the essays which he has 
published, I esteem him an excellent scholar and fine mathematician'." 

" The views of the author are presented in the form of an allegory. He 
conceives himself on the Sun, where he surveys the universe ; he pictures to> 
himself the aspects, relations and conditions of things totally different from 
what they seem on the Earth ; and from his exalted standpoint he criticises 
the theories of Gravitation, Central Forces, Projectiles, Hypothetical Astron- 
omy, Atomic Chemistry, Electricity, Galvanism and Magnetism ; he advances 
some new notions of Matter, and proclaims the seeming paradox that the 
Heat we feel in the sunlight is not emitted to us from the sun, and that we 
never can obtain any knowledge of Cause. He promises, at some near future 
time, to dissipate the mists of Metaphysics and to solve some Knotty ques- 
tions in Ethics. The work is original in conception, comprehensive in plan r 
bold in character, vigorous in style, exhaustive in research, forcible in argu- 
ment, poetical in diction, and erudite in treatment It challenges the scrutiny 
of savans ; it is readable and instructive, and will repay many perusals. We 
are anxious to see the continuance of it, as we regard it as a wonderful 
work, and we esteem it as a valuable acquisition to our stock of scientific- 
lore." — Reviewer. ^ 

Address JAMES S. BURNTON, Publisher, 

149 Grand Street, New York. 



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